Sure! Let's simplify the given expression step-by-step.
We start with the expression:
[tex]\[ 4^{\frac{1}{3}} \cdot 4^{\frac{1}{5}} \][/tex]
1. Use the product of powers property:
This property states that when you multiply two powers with the same base, you add the exponents:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
Here, our base is 4, and the exponents are \(\frac{1}{3}\) and \(\frac{1}{5}\).
So, we add the exponents:
[tex]\[ \frac{1}{3} + \frac{1}{5} \][/tex]
2. Find a common denominator:
To add \(\frac{1}{3}\) and \(\frac{1}{5}\), we find a common denominator, which is 15:
[tex]\[ \frac{1}{3} = \frac{5}{15} \][/tex]
[tex]\[ \frac{1}{5} = \frac{3}{15} \][/tex]
Adding these gives us:
[tex]\[ \frac{5}{15} + \frac{3}{15} = \frac{8}{15} \][/tex]
3. Rewrite the expression:
Now that we have the sum of the exponents, we can rewrite the original expression as:
[tex]\[ 4^{\frac{8}{15}} \][/tex]
4. Simplify the expression:
Finally, we evaluate the power:
[tex]\[ 4^{\frac{8}{15}} \approx 2.0945882456412535 \][/tex]
Therefore, the simplified result of the expression \( 4^{\frac{1}{3}} \cdot 4^{\frac{1}{5}} \) is approximately:
[tex]\[ 2.0945882456412535 \][/tex]
